Hyperbolic space

In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point. Parallel lines are not uniquely paired: given a line and a point not on that line, any number of lines can be drawn through the point which are coplanar with the first and do not intersect it. This contrasts with Euclidean geometry, where parallel lines are a unique pair, and spherical geometry, where parallel lines do not exist, as all lines (which are great circles) cross each other. Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball, rather than polynomially.

Contents

Formal definition

Hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry. It can be thought of as the negative-curvature analogue of the n-sphere. Although hyperbolic space Hn is diffeomorphic to Rn its negative-curvature metric gives it very different geometric properties.

Hyperbolic 2-space, H², is also called the hyperbolic plane.

Models of hyperbolic space

Hyperbolic space, developed independently by Lobachevsky and Bolyai, is a geometrical space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):

It is then a theorem that there are in fact infinitely many such lines through P. Note that this axiom still does not uniquely characterize the hyperbolic plane uniquely up to isometry; there is an extra constant, the curvature K<0, which must be specified. However, it does uniquely characterize it up to homothety, meaning up to bijections which only change the notion of distance by an overall constant. By choosing an appropriate length-scale, one can thus assume, without loss of generality, that K=-1.

Hyperbolic spaces are constructed in order to model such a modification of Euclidean geometry. In particular, the existence of model spaces implies that the parallel postulate is logically independent of the other axioms of Euclidean geometry.

There are several important models of hyperbolic space: the Klein model, the hyperboloid model, and the Poincaré model. These all model the same geometry in the sense that any two of them can be related by a transformation which preserves all the geometrical properties of the space. They are isometric.

The hyperboloid model

Main article: Hyperboloid model

The first model realizes hyperbolic space as a hyperboloid in Rn+1 = {(x0,...,xn)|xi∈R, i=0,1,...,n}. The hyperboloid is the locus Hn of points whose coordinates satisfy

x_0^2-x_1^2-\ldots-x_n^2=1,\quad x_0>0.

In this model a "line" (or geodesic) is the curve cut out by intersecting Hn with a plane through the origin in Rn+1.

The hyperboloid model is closely related to the geometry of Minkowski space. The quadratic form

Q(x) = x_0^2 - x_1^2 - x_2^2 - \cdots - x_n^2

which defines the hyperboloid polarizes to give the bilinear form B defined by

B(x,y) = (Q(x%2By)-Q(x)-Q(y))/2=x_0y_0 - x_1y_1 - \cdots - x_ny_n.

The space Rn+1, equipped with the bilinear form B is an (n+1)-dimensional Minkowski space Rn,1.

From this perspective, one can associate a notion of distance to the hyperboloid model, by defining[1] the distance between two points x and y on H to be

d(x, y) = \operatorname{arccosh}\, B(x,y).

This function satisfies the axioms of a metric space. Moreover, it is preserved by the action of the Lorentz group on Rn,1. Hence the Lorentz group acts as a transformation group of isometries on Hn.

The Klein model

Main article: Klein model

An alternative model of hyperbolic geometry is on a certain domain in projective space. The Minkowski quadratic form Q defines a subset Un ⊂ RPn given as the locus of points for which Q(x) > 0 in the homogeneous coordinates x. The domain Un is the Klein model of hyperbolic space.

The lines of this model are the open line segments of the ambient projective space which lie in Un. The distance between two points x and y in Un is defined by

d(x, y) = \operatorname{arccosh}\left(\frac{B(x,y)}{\sqrt{Q(x)Q(y)}}\right).

Note that this is well-defined on projective space, since the ratio under the inverse hyperbolic cosine is homogeneous of degree 0.

This model is related to the hyperboloid model as follows. Each point x ∈ Un corresponds to a line Lx through the origin in Rn+1, by the definition of projective space. This line intersects the hyperboloid Hn in a unique point. Conversely, through any point on Hn, there passes a unique line through the origin (which is a point in the projective space). This correspondence defines a bijection between Un and Hn. It is an isometry since evaluating d(x,y) along Q(x) = Q(y) = 1 reproduces the definition of the distance given for the hyperboloid model.

The Poincaré models

Main articles: Poincaré disc model, Poincaré half-plane model

Another closely related pair of models of hyperbolic geometry are the Poincaré ball and Poincaré half-space models. The ball model comes from a stereographic projection of the hyperboloid in Rn+1 onto the hyperplane {x0 = 0}. In detail, let S be the point in Rn,1 with coordinates (-1,0,0,...,0): the South pole for the stereographic projection. For each point P on the hyperboloid Hn, let P* be the unique point of intersection of the line SP with the plane {x0 = 0}. This establishes a bijective mapping of Hn into the unit ball

B^n = \{(x_1,\ldots,x_n) | x_1^2%2B\ldots%2Bx_n^2 < 1\}

in the plane {x0 = 0}.

The geodesics in this model are semicircles which are perpendicular to the boundary sphere of Bn. Isometries of the ball are generated by spherical inversion in hyperspheres perpendicular to the boundary.

The half-space model results from applying an inversion in a point of the boundary of Bn. This sends circles to circles and lines, and is moreover a conformal transformation. Consequently the geodesics of the half-space model are lines and circles perpendicular to the boundary hyperplane.

Hyperbolic manifolds

Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold M of constant negative curvature −1, which is to say, a hyperbolic manifold, is Hn. Thus, every such M can be written as Hn/Γ where Γ is a torsion-free discrete group of isometries on Hn. That is, Γ is a lattice in SO+(n,1).

Riemann surfaces

Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group π1=Γ; the groups that arise this way are known as Fuchsian groups. The quotient space H²/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply connected and noncompact. It is the universal cover of the other hyperbolic surfaces.

The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.

See also

References

  1. ^ Note the similarity with the chordal metric on a sphere, which uses trigonometric instead of hyperbolic functions.